Abstract
Closed-form expressions for generalized entropy rates of Markov chains are obtained through pertinent averaging. First, the rates are expressed in terms of Perron-Frobenius eigenvalues of perturbations of the transition matrices. This leads to a classification of generalized entropy functionals into five exclusive types. Then, a weighted expression is obtained in which the associated Perron-Frobenius eigenvectors play the same role as the stationary distribution in the well-known weighted expression of Shannon entropy rate. Finally, all terms are shown to bear a meaning in terms of dynamics of an auxiliary absorbing Markov chain through the notion of quasi-limit distribution. Illustration of important properties of the involved spectral elements is provided through application to binary Markov chains.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amari SI, Nagaoka H (2007) Methods of information geometry (Vol. 191), American Mathematical Soc.
Basseville M (2013) Divergence measures for statistical data processing. J Signal Proc 93:621–633
Beck C, Schögl F (1995) Thermodynamics of chaotic systems: an introduction (No. 4). Cambridge University Press, Cambridge
Ciuperca G, Girardin V (2007) Estimation of the entropy rate of a countable Markov chain. Comm Stat Th Meth 36:2543–2557
Ciuperca G, Girardin V, Lhote L (2011) Computation of generalized entropy rates. Application and estimation for countable Markov chains. IEEE Trans Info Th 57:4026–4034
Cover L, Thomas J (1991) Elements of information theory. Wiley series in telecommunications, New-York
Ekroot L, Cover TM (1993) The entropy of Markov trajectories. IEEE Trans Info Th 39:1418–1421
Darroch JN, Seneta EE (1965) On quasi-stationary distributions in discrete-time finite Markov chains. J App Probab 2:88–100
Gerchak Y (1981) Maximal entropy of Markov chains with common steady-states probabilities. J Oper Res Soc 32:233–234
Girardin V (2004) Entropy maximization for Markov and semi-Markov processes. Meth Comp App Prob 6:109–127
Girardin V (2005) On the different extensions of the Ergodic Theorem of information theory. In: Baeza-Yates R, Glaz J, Gzyl H, Hüsler J, Palacios JL (eds) Recent advances in applied probability. Springer, San Francisco, pp 163–179
Girardin V, Lhote L (2015) Rescaling entropy and divergence rates. IEEE Trans Info Th 61:5868–5882
Girardin V, Regnault P (2016) Escort distributions minimizing the Kullback–Leibler divergence for a large deviations principle and tests of entropy level. Ann Inst Stat Math 68:439–468
Gosselin F (2001) Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology. Adv App Prob 11:261–284
HohoÈldt T, Justesen J (1984) Maxentropic Markov chains. IEEE Trans Info Th 30:665–667
Huillet T (2009) Random walks pertaining to a class of deterministic weighted graphs, article id. J Physics A 42:275001
Kafsi M, Grossglauser M, Thiran P (2015) Traveling salesman in reverse: Conditional Markov entropy for trajectory segmentation. IEEE Int Conf Data Min 2015:201–210
Lambert A (2008) Population dynamics and random genealogies. Stoch Models 24:45–163
Ledoux J, Rubino G, Sericola B (1994) Exact aggregation of absorbing Markov processes using quasi-stationary distribution. J App Prob 31:626–634
Meyer CD (2000) Matrix analysis and applied linear algebra, SIAM Philadelphia
Pronzato L, Wynn HP, Zhigljavsky AA (1997) Using Renyi entropies to measure uncertainty in search problems. In: Mathematics of stochastic manufacturing systems: AMS-SIAM summer seminar in applied mathematics 33. Williamsburg, USA, pp 253–268
Rached Z, Alajaji F, Campbell LL (2001) Rényi’s divergence and entropy rates for finite alphabet Markov sources. IEEE Trans Info Th 47:1553–1561
Rényi A (1961) On measures of entropy and information. In: Proceedings of the fourth Berkeley symposium on mathematical statistics and probability volume 1: contributions to the theory of statistics, The Regents of the University of California
Regnault P, Girardin V, Lhote L (2017) Escort distributions and the Rényi entropy rates of Markov chains, Geometric science of information, Paris
Saerens M, Achbany Y, Fouss F, Yen L (2009) Randomized shortest-path problems: two related models. Neural Comp 21:2363–2404
Menéndez ML, Morales D, Pardo L, Salicrú M (1997) (h,φ)-entropy differential metric. Appl Math 42:81–98
R Core Team (2018) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/
Seneta E (2006) Non-negative matrices and Markov chains, No. 21. Springer Series in Statistics, New York
Sgarro A (1978) An informational divergence geometry for stochastic matrices. Calcolo 15:41–49
Shannon C (1948) A mathematical theory of communication. Bell Syst Techn J 27:379–423, 623–656
Seneta E, Vere-Jones D (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J App Prob 3:403–434
Vallée B (2001) Dynamical sources in information theory: Fundamental intervals and word prefixes. Algorithmica 29:262–306
Varma RS (1966) Generalizations of Rényi’s entropy of order α. J Math Sc 1:34–48
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Girardin, V., Lhote, L. & Regnault, P. Different Closed-Form Expressions for Generalized Entropy Rates of Markov Chains. Methodol Comput Appl Probab 21, 1431–1452 (2019). https://doi.org/10.1007/s11009-018-9679-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-018-9679-3